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"Copula"
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Asymptotics of empirical copula processes under non-restrictive smoothness assumptions
Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.
Journal Article
The grammar of copulas across languages
This volume presents a crosslinguistic survey of the current theoretical debates around copular constructions from a generative perspective. Following an introduction to the main questions surrounding the analysis and categorization of copulas, the chapters address a range of key topics including the existence of more than one copular form in certain languages, the factors determining the presence or absence of a copula, and the morphology of copular forms. The team of expert contributors present new theoretical proposals regarding the formal mechanisms behind the behaviour and patterns observed in copulas in a wide range of typologically diverse languages, including Czech, French, Korean, and languages from the Dene and Bantu families. Their findings have implications beyond the study of copulas and shed more light on issues such as agreement relations, the nature of grammatical categories, and nominal predicates in syntax and semantics.
Book
New copula families and mixing properties
We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie–Gumbel–Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate ψ-mixing Markov chains. Some general results on ψ-mixing are given. The Spearman’s correlation ρS and Kendall’s τ are provided for the created copula families. Some general remarks are provided for ρS and τ. A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.
Journal Article
Multivariate copula and co-copula on BL-algebra
This paper studies the functions of copula on BL-algebra on n-dimension. Our main idea for this study focuses on linking some statistical concepts with BL-algebra. We have proposed several concepts such as some definitions, theories, characteristics of these functions and test their terms with respect to algebra. In fact, our work focuses on determining the type of generalization of each classical copula type in such a forced system. The conditions of copula, co-copula and its related properties have been demonstrated. In the end, there are many different examples that have been shown on the definitions of each type of copula we have built on BL-algebra.
Journal Article
The t Copula and Related Copulas
The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively. /// Dans cet article on décrit les propriétés de la copule t, avec particulière attention envers la dépendance des valeurs extrêmes. Exploitant la représentation de la loi multivariée t par un mélange de Gaussiennes, on construit deux nouveaux types de copule: une version biaisée (skewed t copula) et une version permettant une majeure hétérogénéité dans la modélisation des observations dépendantes (grouped t copula). Deux autres types de copule sont ensuite construits à l'aide de la théorie des valeurs extrêmes. L'une est la copule limite de la loi des maxima de chaque composante d'un vecteur aléatoire avec distribution t (t extreme value copula), l'autre est la copule limite des observations d'un vecteur bivarié obéissant à une loi t, conditionnées a être en dessous d'un certain seuil commun, qu'on baisse progressivement (t lower tail copula). En ce qui concerne les applications pratiques, ces deux dernières copules peuvent être approximées par d'autres copules plus simples et connues, comme celle de Gumbel et celle de Clayton.
Journal Article
Truncated regular vines in high dimensions with application to financial data
Using only bivariate copulas as building blocks, regular vine copulas constitute a flexible class of high-dimensional dependency models. However, the flexibility comes along with an exponentially increasing complexity in larger dimensions. In order to counteract this problem, we propose using statistical model selection techniques to either truncate or simplify a regular vine copula. As a special case, we consider the simplification of a canonical vine copula using a multivariate copula as previously treated by Heinen & Valdesogo (2009) and Valdesogo (2009). We validate the proposed approaches by extensive simulation studies and use them to investigate a 19-dimensional financial data set of Norwegian and international market variables. En utilisant uniquement des copules bidimensionnelles comme unités de base, les copules en arborescence régulière constituent une classe flexible pour modéliser la dépendance pour les grandes dimensions. Toutefois, en grandes dimensions, la flexibilité s'accompagne d'une croissance exponentielle de la complexité. Pour contrecarrer ce problème, nous proposons l'utilisation des techniques de sélection de modèles statistiques afin de tronquer ou encore de simplifier la copule en arborescence régulière. Comme cas particulier, nous considérons la simplification de la copule en arborescence canonique par l'utilisation d'une copule multidimensionnelle telle que présentée dans Heinen et Valdesogo (2009) et Valdesogo (2009). Nous validons les approches proposées par de vastes études de simulation et nous les utilisons pour analyser un jeu de données financières de dimension 19 sur des variables des marchés norvégien et internationaux.
Journal Article
Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation
Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.
Journal Article
Stock market returns and oil price shocks: A CoVaR analysis based on dynamic vine copula models
Crude oil plays a significant role in economic developments in the world. Understanding the relationship between oil price changes and stock market returns helps to improve portfolio strategies and risk positions. Kilian (Am Econ Rev 99(3): 1053–1069, 2009) proposes to decompose the oil price into three types of oil price shocks by using a structural vector autoregression model. This paper investigates the dynamic, nonlinear dependence and risk spillover effects between BRICS stock returns and the different types of oil price shocks using an appropriate multivariate and dynamic copula model. Risk is measured using the conditional value at risk, conditioning on one or more simultaneous oil and stock market shocks. For this purpose, a D-vine-based quantile regression model and the GAS copula model are combined. Our results show, inter alia, that the early stages of the Covid-19 crisis lead to increasing risk levels in the BRICS stock markets except for the Chinese one, which has recovered quickly and therefore shows no changes in the risk level.
Journal Article
High dimensional semiparametric latent graphical model for mixed data
We propose a semiparametric latent Gaussian copula model for modelling mixed multivariate data, which contain a combination of both continuous and binary variables. The model assumes that the observed binary variables are obtained by dichotomizing latent variables that satisfy the Gaussian copula distribution. The goal is to infer the conditional independence relationship between the latent random variables, based on the observed mixed data. Our work has two main contributions: we propose a unified rank-based approach to estimate the correlation matrix of latent variables; we establish the concentration inequality of the proposed rank-based estimator. Consequently, our methods achieve the same rates of convergence for precision matrix estimation and graph recovery, as if the latent variables were observed. The methods proposed are numerically assessed through extensive simulation studies, and real data analysis.
Journal Article